where \(\alpha _j^{t}\)  denotes weights for each data modal, i.e., prior probability for BFs and PAs, \(\hat{p}_i^{t}(\mathbf {z}|j)\)  is the empirical class conditional probability distribution, and \(\mathbf {\mu }_j^{t}\) , \(\mathbf {\Sigma }_j^{t}\)  denote the mean and co-variance for each component, respectively. Estimating the GMM parameters is usually performed through expectation maximization (EM) [1]}, which can be a computationally expensive procedure. However, since we have access to labels of data points, we can decouple the GMM components and compute the GMM parameters for each component independently, using MAP estimates. Consider \(\mathbf {S}_j^{t}\)  to be the support set for BFs (\(j=0\) ) and PAs (\(j=1\) ) in the training dataset, i.e., \(\mathbf {S}_j^{t}=\lbrace \mathbf {x}_i^t\in \mathbf {X}^{t}| \arg \max _c \hat{p}_i^{t}(\mathbf {x}_i^t|c)=j\rbrace \) . Then, we can simply estimate the GMM parameters as:
\(\small \begin{split}&\hat{\alpha }_j^{t} = \frac{|\mathbf {S}_j^{t}|}{N}, \qquad \hat{\mathbf {\mu }}_j^{t} = \frac{1}{|\mathbf {S}_j^{t}|}\sum _{\mathbf {x}_i^t\in \mathbf {S}_j^{t}}\phi ^t_v(\mathbf {x}_i^t), \\& \hat{\mathbf {\Sigma }}_j^{t} =\frac{1}{|\mathbf {S}_j^{t}|}\sum _{\mathbf {x}_i^t\in \mathbf {S}_j^{t}}\big (\phi ^t_v(\mathbf {x}_i^t)-\hat{\mathbf {\mu }}_j^{t}\big )^\top \big (\phi ^t_v(\mathbf {x}_i^t)-\hat{\mathbf {\mu }}_j^{t}\big ).\end{split}\) 
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